Answer :
Certainly! To determine the value of [tex]\( f(-3) \)[/tex] when dividing a polynomial [tex]\( f(x) \)[/tex] by [tex]\( x+3 \)[/tex], we can use the Remainder Theorem. This theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x-c \)[/tex], the remainder of the division is [tex]\( f(c) \)[/tex].
Here's a step-by-step explanation:
1. Identify the divisor and the value to evaluate the polynomial at:
- The divisor is [tex]\( x+3 \)[/tex], which can be rewritten as [tex]\( x - (-3) \)[/tex].
- According to the Remainder Theorem, we will evaluate the polynomial at [tex]\( x = -3 \)[/tex].
2. Synthetic Division Insight:
- Synthetic division is a method used to divide polynomials when the divisor is of the form [tex]\( x-c \)[/tex].
- When you perform synthetic division with [tex]\( x+3 \)[/tex], you're essentially evaluating [tex]\( f(-3) \)[/tex].
3. Remainder Understanding:
- During synthetic division, the remainder is what you get after the division is complete, and this remainder is exactly the value of the polynomial evaluated at [tex]\( c \)[/tex]. In our case, [tex]\( f(-3) \)[/tex].
4. Result:
- Based on our understanding of synthetic division and the Remainder Theorem, the value of [tex]\( f(-3) \)[/tex] in this specific case is [tex]\( 2 \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 2 \)[/tex].
Here's a step-by-step explanation:
1. Identify the divisor and the value to evaluate the polynomial at:
- The divisor is [tex]\( x+3 \)[/tex], which can be rewritten as [tex]\( x - (-3) \)[/tex].
- According to the Remainder Theorem, we will evaluate the polynomial at [tex]\( x = -3 \)[/tex].
2. Synthetic Division Insight:
- Synthetic division is a method used to divide polynomials when the divisor is of the form [tex]\( x-c \)[/tex].
- When you perform synthetic division with [tex]\( x+3 \)[/tex], you're essentially evaluating [tex]\( f(-3) \)[/tex].
3. Remainder Understanding:
- During synthetic division, the remainder is what you get after the division is complete, and this remainder is exactly the value of the polynomial evaluated at [tex]\( c \)[/tex]. In our case, [tex]\( f(-3) \)[/tex].
4. Result:
- Based on our understanding of synthetic division and the Remainder Theorem, the value of [tex]\( f(-3) \)[/tex] in this specific case is [tex]\( 2 \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 2 \)[/tex].
